Log-Cauchy distribution

In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution. In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If X is a random variable with a Cauchy distribution, then Y = exp(X) has a log-Cauchy distribution; likewise, if Y has a log-Cauchy distribution, then X = log(Y) has a Cauchy distribution. The log-Cauchy distribution has the probability density function: where μ {displaystyle mu } is a real number and σ > 0 {displaystyle sigma >0} . If σ {displaystyle sigma } is known, the scale parameter is e μ {displaystyle e^{mu }} . μ {displaystyle mu } and σ {displaystyle sigma } correspond to the location parameter and scale parameter of the associated Cauchy distribution. Some authors define μ {displaystyle mu } and σ {displaystyle sigma } as the location and scale parameters, respectively, of the log-Cauchy distribution. For μ = 0 {displaystyle mu =0} and σ = 1 {displaystyle sigma =1} , corresponding to a standard Cauchy distribution, the probability density function reduces to: The cumulative distribution function (cdf) when μ = 0 {displaystyle mu =0} and σ = 1 {displaystyle sigma =1} is: The survival function when μ = 0 {displaystyle mu =0} and σ = 1 {displaystyle sigma =1} is: The hazard rate when μ = 0 {displaystyle mu =0} and σ = 1 {displaystyle sigma =1} is: The hazard rate decreases at the beginning and at the end of the distribution, but there may be an interval over which the hazard rate increases. The log-Cauchy distribution is an example of a heavy-tailed distribution. Some authors regard it as a 'super-heavy tailed' distribution, because it has a heavier tail than a Pareto distribution-type heavy tail, i.e., it has a logarithmically decaying tail. As with the Cauchy distribution, none of the non-trivial moments of the log-Cauchy distribution are finite. The mean is a moment so the log-Cauchy distribution does not have a defined mean or standard deviation.